\section{Project Overview}
\label{sec:overview}

The repository organises the locELM implementation as a standalone Python package accompanied by comprehensive documentation and experiment artefacts. The overall directory layout, excerpted from the documentation index and final report, is reproduced in Figure~\ref{fig:layout}.

\begin{figure}[h]
  \centering
  \begin{minipage}{0.9\textwidth}
  \begin{verbatim}
reproduct-paper/
|-- locELM/                      # Core package (~2,400 LOC)
|   |-- core/                    # Domain, networks, continuity
|   |-- solvers/                 # Linear and nonlinear PDE solvers
|   |-- utils/                   # Error metrics and plotting
|   `-- examples/                # 1D and 2D Helmholtz scripts
|-- experiments/                 # Empirical results and reports
|-- README.md, IMPLEMENTATION.md # User and developer documentation
|-- FINAL_REPORT.md              # Project-level synthesis
|-- BUG_FIX_SUMMARY.md           # Debug investigation notes
`-- requirements.txt, setup.py   # Installation metadata
  \end{verbatim}
  \end{minipage}
  \caption{Repository structure highlighting code, experiments, and documentation.}
  \label{fig:layout}
\end{figure}

The package targets reproducibility of the paper's linear Helmholtz experiments and provides the following deliverables:
\begin{itemize}
  \item A modular Python codebase covering domain decomposition, neural architectures, continuity enforcement, and both linear and nonlinear solvers.
  \item Executable examples for one- and two-dimensional Helmholtz equations with plotting utilities.
  \item Detailed documentation: user-oriented guidance, implementation notes, bug investigations, and experiment summaries.
  \item Recorded experiment outputs and markdown reports that capture observed accuracy, timing, and outstanding issues.
\end{itemize}

Together these assets establish a solid baseline implementation while surfacing the remaining discrepancies that must be addressed to match the reference study.

\begin{figure}[h]
  \centering
  \includegraphics[width=\textwidth]{figures/locelm_structure.png}
  \caption{Module-level locELM architecture showing how core components feed the solvers, utilities, and example scripts.}
  \label{fig:locelm-structure}
\end{figure}
